ME 306 – Fluid Mechanics II
Boundary Layer Theory
- Part 3 Dr. Merve Erdal
Mechanical Engineering Department
Middle East Technical University
(Office: B-304, ph:210-5237, merdal@metu.edu.tr)
1
Blasius’ Exact Solution of B/L Eqn.s for
Flow Parallel to a Flat Plate
Boundary conditions:
At ! = 0
$ = % = 0 (no slip, no flow across)
As ! → ∞
$ → ( (patching w/ free stream)
• Blasius was Prandtl’s student
• He came up with the change of variables,
,
defined stream function, / = 0-$ &/' %(")
that satisfies continuity for ! =
()
,&
(*
=−
()
(%
and showed that the dimensionless velocity profile ! in B/L is a function of the single
"# &/'
!
composite variable ", ! = $% " and the velocity &, & =
["% ! " − % " ]
$%
+,
From the freestream inviscid flow solution (using Bernoulli equation),
= 0 since $
+remains constant. Substitution of all in B/L equations:
0
single ordinary differential equation!
coupled partial differential equations
with boundary conditions:
At ! = 0
$ 0 = $′(0) = 0 (no slip,no flow across)
2
As ! → ∞
$′(∞) → 1
(patching)
Blasius’ Exact Solution (Cont.’d)
• Solution obtained for !(#) by numerical integration of the governing ODE
• From the knowledge of !(#), every flow parameter can be obtained
Velocity profile:
=3
=4′(3)
=3
=4′(3)
B/L thickness % found from:
,
= 0.99
This happens when # ≈ 5.0 (see table). Rearranging # = 5.0 with & = %,
& = % when
where Re7 =
(note %~ 8)
-8
9
Results apply to ALL laminar, steady,
incompressible, 2-D flows over a flat plate
(How does the “flatness” of plate and its
orientation simplify the flow?)
3
Blasius’ Exact Solution (Cont.’d)
• From their integral-form definitions, B/L displacement and momentum thicknesses:
!" 1.721
=
#
Re+
!, 0.664
=
#
Re+
Re+ =
6#
F
How does the wall shear
stress change along the flow?
• The wall shear stress at the plate position #:
34
:;
@ ;/6
:B
# =8 9
= 86
9
9
= 86D EE 0
:< =>?
@B C>? :< =>?
6 G/7
0.332 56 7
→ 34 # =
F#
Re+
J: plate width
K: plate length
• Local skin friction coefficient, 01,+ :
34 #
0.664
=
=
56 7
Re+
2
01,+
• Drag force N # on plate up until position # :
+
N # = P 34 # J@# =
?
0.66456 7 J#
Re+
How does the drag force change
with increased plate length?
• Drag coefficient for the whole plate length, K: 0L,M
N K
1.328
=
=
56 7
ReM
JK
2
4
To Sum Up:
Blasius’ exact solution results are valid for laminar, steady,
incompressible, 2-D, high !" flows over a flat plate. The consequent
approximations (for general B/L and specific flat plate flows) were:
∗
• Re! < !""#
(laminar flow) (Exact solution NOT valid if turbulent)
&' &'
•
≪
→ ' = ' * only (no change in across B/L).
&( &*
Outside B/L, inviscid flow (justified). From Bernoulli,
./(*)%
9'
9/
' * +
= constant →
+ ./
=0
2
9*
9*
9/
1 9'
→/
=−
(* − component of Euler eqn!)
9*
. 9*
For flat plate, 9/ = 0 → 9' = 0 → ' = constant (Exact solution NOT
!"
valid
if
≠ ")
9*
9*
!#
• Re& ≫ 1 for B/L (thin layer) assumptionà B/L solutions not valid
near the leading edge
*#$$% = 5×10& undisturbed flow over for smooth plate
5
Example
Laminar B/L formed on one side of a plate of length ! in parallel
to a uniform flow produces a drag ". How much must the plate
be shortened to reduce the drag to "/4?
ℓ
!
Drag: "/4
Drag: "
" ! =
0.664()* +!
Re.
If drag is reduced to "/4,
=
0.664()* +!
)!
/
→" ! ~ !
"3
"
!
5
=
=
→ℓ=
"* "/4
67
ℓ
B/L thinner near leading edge à velocity gradient on wall higher à 89
higher à Drag force on the plate not uniformly distributed (higher near
the front). That’s why such a high reduction in plate length needed.
6
Turbulent B/L flow
$%
•
Newton’s Law of Viscosity: ! = #
NOT valid in
$&
turbulent flow
•
No exact theory for turbulent flat-plate flow
•
Empirical models exist for turbulent shear stress
•
Many numerical solutions of B/L equations with
empirical stress models
•
Turbulent velocity profile very different from
laminar profile due to high level of mixing
(experimentally obtained for
parallel flow over flat pate)
•
Turbulent
profiles
flatter but
there is still
no-slip
à have a larger velocity gradient at the wall.
When flow first becomes turbulent over flat
plate à sharp rise in wall shear stress
•
High level of mixing à larger boundary layer
thicknesses than laminar profiles
7
More on Turbulent Boundary Layer
•
Velocity at any given location in the flow is unsteady in a random fashion.
•
Flow can be thought of as a jumbled mix of interwined eddies (or swirls) of different
sizes (diameters and angular velocities).
Fluid quantities (mass, momentum, energy) are
Time-averaged
convected downstream in the freestream
velocity profile, !'
direction (as in a laminar boundary layer) but also
! ", $, % = !' + !) (%)
across the boundary layer in the direction
perpendicular to the plate (unlike a laminar
boundary layer) by the random transport of finite!
sized fluid particles associated with the turbulent
eddies.
Turbulent eddies
•
•
Cross-stream mass transport: Very little net mass
transfer across boundary layer (dominating flow
rate parallel to plate) though plenty of random
motion of fluid particles perpendicular to the plate
•
Cross-stream momentum transport :
•
In laminar flow: at molecular level due to viscous diffusion
•
In turbulent flow: at macrolevel by convection
8
Von Karman’s Momentum Integral Equation
• Von Karman was another student of Prandtl
• Developed an alternative but approximate (not exact) solution to the
boundary layer equations by approximating the velocity profile in the
boundary layer
• Performed a control volume analysis inside the boundary layer to obtain a
relation for wall shear stress in integral form
• Results applicable to turbulent flow as well as laminar flow!
+,
≠ 0 flows (with no separation)
• Results also applicable to
+Consider the shown
infinitesimal (*+ long)
control volume inside
the B/L for a steady,
incompressible, 2D flow.
a free streamline
edge of B/L
1
2
Continuity: '̇ &'(% + '̇ %)* = '̇ !"#$%
and '̇ !"#$%
,̇ $+,
,̇ !"#$
3
CV
(unit width)
3+d3
02
&
,̇ '()*$
,̇ !"#$ = - . /01
solid boundary
%
,
* '̇ &'(%
* '̇ &'(%
*
= '̇ &'(% +
*+ → '̇ %)* =
*+ → '̇ %)* =
- . /*0 *+
*+
*+
*+
+
9
Von Karman’s Momentum Integral Equation (Cont.’d)
• Next, the x component of the momentum equation in integral form:
Σ2- = −4̇ -,&'(% − 4̇ -,%)* + 4̇ -,!"#$% (momentum rates on left and right CS’E OUT from CS)
where
1 *J
*J
*J
Σ2- = JK + J +
*+ *K − JK + J*K +
K*+ − N1 *+ = − K*+ − N1 *+
2 *+
*+
*+
≈*0,
4̇ -,!"#$%
* 4̇ -,&'(%
* 4̇ -,&'(%
= 4̇ -,&'(% +
*+ → −4̇ -,&'(% + 4̇ -,!"#$% =
*+
*+
*+
,
,
4̇ -,&'(% = - . /2 *0 → −4̇ -,&'(% + 4̇ -,!"#$% =
+
*
- . /2 *0 *+
*+
+
,
4̇ -,%)*
*
= '̇ %)* O + = O +
- . /*0 *+
*+
43
2
3
CV
free stream velocity
9(2)
1 04
4+
02 03
2 02
1
+
projected area of top
CS on plane normal to )
mean pressure
on top CS
3+d3
02
8- 02
4+
04
02
02
3 + 03 ≈ 43 + 403 +
04
302
02
10
Von Karman’s Momentum Integral Equation (Cont.’d)
• Putting all together in the momentum equation:
=
Σ"# = −
=
&'
&
&
)&( − *+ &( = −-̇ #,/012 − -̇ #,234 + -̇ #,67892 =
: ; >? &@ &( − A (
: ; >&@ &(
&(
&(
&(
<
=
→−
<
=
&'
&
&
) − *+ =
: ; > ? &@ − A (
: ; >&@
&(
&(
&(
<
• In the free stream: A (
<
&A (
1 &'
&'
&A (
=−
→
= −:A (
&(
: &(
&(
&(
• Then,
=
=
=
&'
&'
&A (
&A (
)=
; &@ = −:A (
; &@ = −: ; A (
&@
&(
&(
&(
&(
<
<
<
• Momentum equation becomes:
=
=
=
&A (
&
&
?
:;A (
&@ − *+ =
: ; > &@ − A (
: ; >&@
&(
&(
&(
<
<
<
11
Von Karman’s Momentum Integral Equation (Cont.’d)
• Rearranging momentum equation:
*
*
*
%
%/ &
%
,
!" = −
' ( + %- + ' ( / &
%- + / &
' ( +%%&
%&
%&
)
)
)
• Note that
*
*
*
%
%/
%
' ( /+%- = ' (
+%- + /
' ( +%%&
%&
%&
)
*
)
)
*
*
%
%
%/
→/
' ( +%- =
' ( /+%- − ' (
+%%&
%&
%&
)
)
)
• Substitute in the momentum equation and rearrange:
*
*
%
%/
!" =
' ( +(/ − +)%- + '
((/ − +)%%&
%&
)
)
Von Karman’s
Momentum
Integral Equation
which can be also be expressed as:
*
*
%
+
+
%/
+
!" =
'/ , (
1−
%- + '/
( 1−
%%&
/
/
%&
/
)
)
Von Karman’s
Momentum
Integral Equation
12
Von Karman’s Momentum Integral Equation (Cont.’d)
Using the definition of boundary layer displacement and momentum
thicknesses, !" and !# , the momentum equation can be reexpressed in this
very compact form:
)* =
"1
,
,.
-!# . / + -!" .
,%
,%
Von Karman’s Momentum
Integral Equation
"4
• If
= 0 (or
= 0) in the flow such as flow parallel to a flat plate, then
"2
"2
.: constant and
7
)* =
-. /
,!#
,
8
8
/
= -.
5
1−
,;
,%
,%
.
.
6
Von Karman’s Momentum Integral
"4
Equation for any flow with = 0
"2
(such as parallel flow over flat plate)
Then, if we know the momentum thickness !# (%), the drag force '(%) on a
flat plate of width ( up to the position % from the leading edge can be found
as
2
' % = 5 )* (,% =
6
2
( 5 -. /
6
,!#
,% = -(. /
,%
7< (2)
5
7< (6)
,!# → > ? = @ABC DE (?)
13
Laminar vs. Turbulent B/L Velocity Profiles
!
%
• Velocity function: = $
"
&
In laminar flow, can approximate
velocity by
!
6
=5
+B
$
7
!
6 '
6
• parabolic: = 5
+9
+C
$
7
7
!
6 '
6 '
6
+9
+;
+D
• cubic: = 5
$
7
7
7
!
6
• sinusoidal: = 5 sin 9 + C
$
7
• linear:
profiles
&/+
For turbulent flow, ! = 6
where
$
7
A = B8
9
7
if FG,- < FG% < 1×10.
if 1×10. < FG% < 1×10/
if 1×10/ < FG% < 1×100
(!":; = 5×10< only IF smooth plate)
14